Uemura, T.
Osaka J. Math.
32 (1995), 861-868
ON WEAK CONVERGENCE OF DIFFUSION
PROCESSES GENERATED BY ENERGY FORMS
TOSHIHIRO UEMURA
(Received February 10, 1994)
0.
Introduction
Convergences of closed forms, energy forms or energy functions have been
studied by many authors (see e.g. [l]-[3], [5]-[8]). It is important to know, given
an energy form, if it can be approximated by "nice" ones, or, given a sequence
of energy forms, what their "limit" is.
In this paper we consider a sequence of forms $\u,v) = \R*(An(x)Vu(x\
Vv(x))dφ2(x)dx with certain domains on L2(Rd\φldx\ where φn are locally bounded
functions on Rd, An are (dxd) symmetric matrix valued functions on Rd, ( , )d
means the inner product on Rd and Vu = ^lu9V2u9 9Vju) is the distributional
(weak) derivative of u. Take strictly positive, bounded functions fn with
\Rdfnφldx=\ and denote by {Xt,Pnx,x e Rd] the diffusion processes associated with
the forms S>n. We study the weak convergence of the probability measures
{JPJJtn,« = l,2, } with dmn=fnφ2ldx, when the date An, φn9 and fn converge a.e. on
Rd, as n -> oo.
Although our main result (see section 1) is similar to that of T.J. Lyons and
T.S. Zhang [5], we assume only a certain local boundedness of φn, while a uniform
boundedness on the whole space is assumed in [5]. In order to obtain the result
in [5], they generalized the theorem of Kato and Simon on monotone sequence
of closed forms (see M. Reed and B. Simon [7]) used by S. Albeverio, R. H0egh-Krohn
and L. Streit [1]. We will instead adopt the Mosco-convergence of closed forms(see
U. Mosco [6]) to prove our theorem.
S. Albeverio, S. Kusuoka and L. Streit [2] obtained a semigroup convergence
by imposing the regularity conditions that there exist R>0 and C>0 such
that, the restrictions of φn to Rd — BR is of class C2 and the growth order of
χ-Vφn/φn is not greater than C\x\2 on Rd — BR. No smoothness on An9 φn is
required in the present approach.
1.
Statement of Theorem
Let φn(x\ φ(x) be measurable functions on Rd and An(x\ A(x) be (dxd) symmetric
d
matrix valued functions on R . Consider the following conditions:
862
T. UEMURA
(A.I) (i) there exists a constant <5>0 such that
0 <-\ξ\2 <(An(x)ξ,ξ)d< δ\ξ\\
o
for dx-a.e.xERd,ξ e Rd,n e N.
(ii) for any relatively compact open set G of Rd, there exist constants λ(G), Λ(G)>0
such that,
0 < λ(G) < φn(x) < Λ(G),
for dx-a.e.x E Gji G N,
(iii) φn(x) -> φ(x\ dx-a.e. on Rd,
(iv) An(x) -> A(x) in matrix norm, dx-a.e. on Rd.
We consider the forms
<ΠM = ^(An(x)Vu(x\Vv(x))dφ2
(x)dx,
^" = {UE L2(Rd;φ2dx): Vμ E L2(Rd φ2ndx\i = 1,2, - - -,</},
for AZ = 1,2,3, •••,
M= I
μWVφ),Vφ))^2Wrfx,
(1.2)
^ = {u E L2(Rd;φ2dx): V,u E L2(Rdψ2dx),i = 1,2, - •-,</}>
Our assumption (A.I) implies that the forms (1.1) and (1.2) are regular local
Dirichlet forms on L2(Rd\φldx) and L2(Rd\φ2dx) (called "energy forms")
respectively. It follows from M. Fukushima, Y. Oshima and M. Takeda [4] that
there exist diffusion processes M" = {Xt,Pnx,x E Rd} and M" = {Xt,Px,x E Rd} associated
with S"1 and $ respectively. Further, we consider the following condition:
(A.2) there exists a constant c>0 such that supn\Brφ2dx<ecr2, for all r>0.
Then by condition (A.2) and Theorem 2.2 in M. Takeda [8], these processes
are conservative. For every relatively compact open set G of Rd, we consider the
Dirichlet forms of part on G associated with (1.1) and (1.2):
'G(M= f
(An(x)Vu(x)yv(x))dφ2(x)dx9
J G
&l-^Hl(G)
(1.3)
2
2
onL (G;φ dx\
WEAK CONVERGENCE OF DIFFUSION PROCESSES
863
for n = 1,2,3, •-,
(A(x)Vu(x\Vυ(x))dφ2(x)dx,
,v)=
Jc
JG
(1.4)
1
^G = H 0(G)
2
2
onL (G'9φ dx),
Now take strictly positive functions/„ of L^(Rd;φ2dx) andfoίL1(Rd 9φ2dx) and
assume the conditions below:
(A.3) (i)
JR*
dmn=
JR<*
dm — \, where dmn=fnφ2dx
(ii)
for any compact set K, sup||/J|Loo(^;^JC)<oo,
(iii)
fn(x) ->/(x), dx-a.e. on Rd.
and dm=fφ2dx,
n
It follows from conditions (A.2), (A.3) and Theorem 3.1 in M. Takeda [8] that
the sequence of probability measures {/>Jtn,«=l,2, } is tight on C([0,oo) -> Rd).
Moreover we can assert as follows:
Theorem. Assume the conditions (A.1)-(A.3).
weakly to Pm on C([0,oo) -> Rd).
Then {Pnmn,n=\,2,-"}
converges
2. Proof of Theorem
In order to carry out the proof of Theorem, we need some lemmas and notations.
Henceforth, for a form (^,^(^)) on a Hubert space ^, we let £'(u,u)= oo for every
u e ffl — 3)(β\ Here a form means a non-negative definite symmetric form on ^,
not necessarily densely defined. As was mentioned in the introduction, we use
the notion of the Mosco-convergence of forms, which is defined as follows:
DEFINITION. A sequence of forms $n on a Hubert space 2? is said to be
Mosco-convergent to a form ^ on Jf if the following conditions are satisfied;
(M.I) for every sequence un weakly convergent to u in ^f,
(M.2) for every u in Jf , there exists un converging to u in Jf , such that
In [6], U. Mosco showed that a sequence of closed forms §n on a Hubert space
864
T. UEMURA
J f is Mosco-convergent to a closed form $ on Jf if and only if the resolvents
associated with <f" converges to the resolvent associated with $ strongly on tf.
In order to use Mosco's theorem, we introduce related forms:
H;dx):u/φn
for « = 1,2,3,-,
(2.2)
By the unitary map f\->φn lf between L2(G;dx) and L2(G;φ2dx) and by the
condition (A.I), the forms J/"'G and stfG are closed on L2(G\dx).
Lemma. Assume the condition (A.I).
to the form $4G on L2(G;dx).
Then the forms stfn'G is Mosco-convergent
Proof. We have to check the conditions (M.I) and (M.2).
First we note that, from the condition (A.I), there exist (dx d) symmetric matrix
valued functions ^/An(x) = (σnij(x)}
following properties:
and ^/A(x) = (σij{x))
defined on Rd with the
(i) An(x) = (^AJ(x))2, A(x) = (^A(x))2,
(ii)
^/An(x) -»• ^/A(x) in matrix norm, dx-a.e. on Rd.
In particular, \^/An(x)ξ\ < ^/δ\ξ\9dx - a.e.x e Rd,ξ e Rd,n e N. Hence σ^x) is
uniformly bounded on Rd and converges to σ^x^dx-a.e. as n -*• oo for each ij.
Proof of (M.I).
2
Suppose un-+u weakly in L (G\dx).
Then we have
+ oo > lim inf j^"
w-»oo
Γ
/
\
liminf |v( — }\2dx,
"^ JG V^./
We may assume
WEAK CONVERGENCE OF DIFFUSION PROCESSES
865
and we can take a subsequence {nk} such that VJ —— ] is weakly convergent to
\ΦnJ
an element Λ t e L (G;dx) for each ί = 1,2, -4 and lim inf jtfn'G(un,un) = lim y4Πk'G(Mπ ,WM ).
2
n-»oo
ί
/
fc->oo
\
converges to u/φ weakly in L (G\dx\ because φn
l
converges to φ~ dx-a.e. on G. This shows that
JG
htfdx=-
k
Γ
VJ — V& = ^Vtfdx,
7 \0»fc/
JG0 W k
2
k
1
and M π /φ π
is uniformly bounded and
for all
-Vtfdx,
JcΦ
Thus we have Λi = V I ( —), /= 1,2,•••,£/, and in particular
\Φ/
Furthermore Σ^.σ^V. ί —^ }φn,
converges
VΦn/
to Σ^iσ . V/l — }φ weakly in
\Φ/
L2(G\dx\ since σ^π is uniformly bounded and converges to σ^φ dx-a.e. on G as
/i -> oo for each i, /
Consequently,
= lim
/ M
AT~"
"
= rlim fI iL/^
__
"^°°JG
\Φnk.
d
d
'U
Σ I I Σ <w ΦProof of (M.2). Let w be in 2(dG\ that is, uεL2(G'4x] and w y .
Accordingly there exists a sequence {ηn} in C^°(G) such that \\u/φ — ηn\\Hι(G) converges
to 0 as n -> oo. Put un = φnηn. Then we can see that un -> u in L2(G\dx). Further,
using again the property of the sequence σ"7φn observed above, we get that
d
Therefore we have
d
/M\
866
T. UEMURA
n G
f ' (u\ wu}n)=
i
JG
\
n
i
.
\ΦJ
I7
Σ II Σ
\ i
\Φn
Φ
q.e.d.
This lemma shows that, if we let Hn'G and HG be the selfadjoint operators
associated with the forms J/ W>G and s$G respectively, then Hn*G converges to HG in the
strong resolvent sense, hence, in the semigroup sense on L2(G\dx) by Mosco's theorem.
Let Hφ'G and HG also denote the selfadjoint operators associated with the forms
<Γ'G and £G respectively. Then by the unitary map /W φ~ lf between L2(G;dx)
and
On the other hand, let Mn G = {Xt9Pnx GjceG} and MG = { Xt9PG,x e G} be the
diffusion processes associated with the forms $n'G and $G respectively. Because
$Π'G is the part of $ " on G as we have already noted, the behaviour of the process
{ Xt,Pnx,x e Rd} is the same as that of {Xt,Pnx-G,xeG} before it leaves G for each n.
Now we can give the proof of Theorem:
Proof of Theorem. By Lemma and the argument following it, we see that
φne~ φ'nφ~ Converges to φe~tHφφ~l strongly on L\G\dx\ Here e~tHn*'? and
e~tH* denotes the semigroups associated with <fn'G and <fG respectively. Therefore,
by virtue of Theorem 7 in [1], Pn^G converges to PG in the finite dimensional
distribution sense.
On the other hand, one has from condition (A.2) and Lemma 2.1 in [8] that
tH
for all
d
Then, for any Q<ti<t2'-<tp, AtE^(R \ /= 1,2, •••,/? and ε>0, there exists
an r > 0 such that supnP"mn(tp > τr) < ε / 2. Moreover, we can see that Pm(tp > τr) < ε / 2.
Here τr denotes the exit time for the open ball Br with radius r and center O.
Let A = {XtιeA1,Xt2eA2,'",XtpeAp}.
Then we see that
- Pnmn(\ n {tp < τr})|
Λ n {tp < τr}) - Pm(\ n {tp < τr})\
+
\Pm(An{lp<τr})-Pm(\)\
WEAK CONVERGENCE OF DIFFUSION PROCESSES
867
Λ n{tp< τr}) - Pm(A n [tp < τr})|.
The first and second term of the right hand side are less than ε. Since the
last term is the finite dimensional distribution of Mn'Br and MBr, we conclude that Pnmn
converges to Pm in the finite dimensional distribution sense.
We have already noted the tightness of {/* J on C([0,oo) -> Rd). Thus the
proof of Theorem is completed.
q.e.d.
Example. Let /be a locally bounded measurable function on Rd9 and consider
a mollifier, e.g., 7'(x) = yexp(— 1 / 1 — |x|2) for \x\<l9j(x) = 0 for |jc|>l, where γ is a
constant to make \Rdj(x)dx = 1 . We put je(x) =j(x / ε) / sd,fε(x) = $Rdjε(x -y)f(y)dy, for
any ε > 0. Since fε converges to / in L2(G;dx) for each relatively compact open set
G, we can take a sequence επ converging to 0 such that/ £M converges to/, dx-a.e. on
Rd. Thus if we set φn(x) = expfEn(x),φ(x) = exp/(x), and assume that there exists a
constant c>0 with $Bre2f(x)dx<eCjr2, for r>0, then φn,φ satisfies the conditions (A.1)
and (A.2). Therefore we have the weak convergence statement for the processes
associated with φmφ and An = A = identity matrix.
ACKNOWLEDGEMENT. The author would like to express his gratitude to
Professor M. Fukushima for helpful advice. The author thanks Professor
M. Takeda, who encourages him to study this subject and gives him valuable
comments. The author is also grateful to Professors N. Jacob, I. Shigekawa and
K. Kuwae for helpful suggestions.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
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S. Albeverio, S. Kusuoka and L. Streit: Convergence of Dirichlet Forms and associated Schrόdinger
operators, J. Funct. Anal., 68 (1986), 130-148
H. Attouch: Variational Convergence for Functions and Operators, Pitman, Boston-LondonMelbourne, 1984
M. Fukushima, Y. Oshima and M. Takeda: Dirichlet Forms and Symmetric Markov Processes,
Walter de Gruyter & Co., Berlin-New York, 1994
T.J. Lyons and T.S. Zhang: Note on convergence of Dirichlet processes, Bull. London Math. Soc.,
25 (1993), 353-356
U. Mosco: Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421
M. Reed and B. Shimon: Functional Analysis (Methods of modern mathematical physics vol.1,
enlarged edition), Academic Press, New York-San Fransisco-London, 1980
M. Takeda: On a martingale method for symmetric diffusion processes and its application, Osaka
J. Math., 26 (1989), 603-623
868
T. UEMURA
Department of Mathematical Science
Faculty of Engineering Science
Osaka University, Toyonaka, Osaka, Japan